# Representation of Rotations

Consider what happens when an object rotates continuously over time, i.e. the rotational matrix is a function $$R(t)$$ of time.

## The derivative

1. Rotation is represented by an orthogonal matrix $$R$$ $R(t)\cdot R^T(t)=I$
2. Implicit derivation $\dot R(t)\cdot R^T(t)+R(t)\cdot\dot R^T(t)=I$
3. by transposing the product and moving one term across, we have $\dot R(t)\cdot R^T(t) = -(\dot R(t)\cdot R^T(t))^T$
4. This is a skew-symmetric matrix, hence $\exists \vec{\omega}\in\mathbb{R}^3, \text{s.t.} \dot R(t)\cdot R^T(t) = \hat\omega(t)$
5. Multiply by $$R(t)$$ to get $\dot R(t) = \hat\omega(t)\cdot R(t)$
6. If $$R(t_0)=I$$ as an initial condition, then $$\dot R(t)=\hat\omega(t)$$

Note $$so(3)$$ is the space of all skew-symmetric matrices.

1. First Order approximation $R(t_0+dt)\approx I + \hat\omega(t_0)dt$

## The differential equation

Let $$x(t)$$ be a point rotated over time.

Assume that $$\omega$$ is constant.

1. ODE: $\dot x(t) = \hat\omega x(t), \quad x(t)\in\mathbb{R}^3$
2. Solution: $x(t) = e^{\hat\omega t} x(0)$
3. where $e^{\hat\omega t} = I + \sum_{i=1}^\infty \frac{(\hat\omega )^i}{i!}$
4. The rotational matrix $R(t)=e^{\hat\omega t}$ signifies a rotation around the axis $$\omega$$ by $$t$$ radians.

$\exp : \mathrm{so}(3)\to\mathrm{SO}(3); \hat\omega\mapsto e^{\hat\omega}$

This is a map from a Lie algebra to a Lie group.

For any $$R$$, such an $$\hat\omega$$ can be found, not necessarily unique.

Rotation is obviously periodic. A rotation by $$2\pi$$ is back to start.

Note Only three degrees of freedom.

## Logarithm

Theorem 2.8 page 27 in the textbook

$R = \begin{bmatrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33} \end{bmatrix} = \exp(\hat\omega)$ where $\DeclareMathOperator{\tr}{trace} ||\omega|| = \cos^{-1}\big(\frac{\tr(R)-1}2\big)$ and

$\frac{\omega}{||\omega||} = \frac1{2\sin(||\omega||)} \begin{bmatrix} r_{32}-r_{23}\\ r_{13}-r_{31}\\ r_{21}-r_{12} \end{bmatrix}$